On page 5 in the book Foundations of Analysis by Landau there is a proof, which I am focusing on a part of and elaborating it to emphasize my point, and possibly uncover my misunderstanding:
... 1 is in this set M, since for $x=1$ there exists a natural number namely $x+y=y'$ which conforms to our definition* for each $y$ since $x+y$ for $y=1$ is $x+1=x'=1'=y'$ and for $y\ne 1$ it is $x+y'=(x+y)'=(1+y)'=\ldots$
But here I need $1+y=y'$ to get to the next necessary step, which I can only conclude using commutativity, since our definition* has $\chi +1=\chi'$ not $1+\chi=\chi'$, but commutativity is only introduced on the next page, and which to my greatest surprise says "by the construction in proof of Theorem 4"! Isn't this circular?
* the recursive formula defining addition using the successor function
Let's use the letter $a$ as a variable in our definitions, and reserve the letter $y$ as a variable in our verifications of the theorem's conclusions. (The author uses $y$ for both, which may be confusing.)
When $x=1$, the author is defining $x+a$ to mean $a'$ for all $a$. In order to investigate $x+y'$, we substitute $y'$ for $a$ in the definition, giving us $x+y'=(y')'$. Now we want to expand the $y'$ inside the parentheses. Substituting $y$ for $a$ in the definition gives us $x+y=y'$, which is just what we need. Putting it all together, $$x+y'=(y')'=(x+y)',$$ which is the desired conclusion.
In your interpretation, you wrote $x+y'=(x+y)'=\cdots$, which assumes what we're trying to prove! Then, you try to prove what we've already assumed, namely $1+y=y'$. That's working backwards, which is sometimes a useful problem-solving strategy, but it can lead you in circles when it comes to subtle matters of definition.